Molecular absorption spectroscopy is a technique that uses the interaction of energy with a molecular species to qualitatively and/or quantitatively study the species, or to study physical processes associated with the species. The interaction of radiation with matter can cause redirection of the radiation and/or transitions between the energy levels of the atoms or molecules. The transition from a lower level to a higher level with an accompanying transfer of energy from the radiation to the atom or molecule is called absorption. When a molecule absorbs light, the incoming energy excites a quantized structure to a higher energy level. The type of excitation depends on the wavelength of the light. Electrons are promoted to higher orbitals by ultraviolet or visible light, vibrations are excited by infrared light, and rotations are excited by microwaves. The infrared (IR) region is generally defined as radiation with wavelength in the range from 1 to 50 μm. Frequency is a measure of the type of radiation related to wavelength such that frequency equals the speed of light divided by the wavelength. A common unit of radiation frequency is cm−1, which is simply the reciprocal of the wavelength expressed in cm. The 0.7 to 2.5 μm wavelength region is generally called the near-infrared (NIR), the 2.5 to 15 μm wavelength region is referred to as the mid-infrared and the 15 to 50 μm wavelength region is called the far-infrared. The wavelengths of IR absorption bands are characteristic of specific types of chemical bonds, and IR spectroscopy finds its greatest utility in the identification of organic and organometallic molecules.
The data that is obtained from spectroscopy is called a spectrum and the instrument or apparatus that produces the spectrum is called a spectrometer. An absorption spectrum shows the absorption of light as a function of its wavelength. The spectrum of a particular atom or molecule depends on its energy level structure. A spectrum can be used to obtain information about atomic and molecular energy levels, molecular geometries, chemical bonds, the interactions of molecules, and related processes. Often, spectra are used to identify the components of a sample (qualitative analysis). Spectra may also be used to measure the amount of material present in a sample (quantitative analysis).
The quantum mechanical derivation of the strength of the absorption begins with the transition moment:R=<Xi|u|Xj>where Xi and Xj are the initial and final states, respectively, and u is the electric dipole moment operator: u=u0+(r−re)du/dr+ . . . , where uo is the permanent dipole moment, which is a constant, r is the radial length of the bond for infrared absorption, and re is the average equilibrium bond length. Because <Xi|Xj>=0 for i≠j according to the laws of quantum mechanics, R simplifies to:R=<Xi|(r−re)du/dr|Xj>
The result is that there must be a change in dipole moment during the vibration of the atoms of a molecule for the molecule to absorb infrared radiation. There is usually no dipole moment change during symmetric stretches of symmetric molecules, so that these bonds usually do not absorb infrared radiation.
Gaseous molecules are found only in discrete states of vibration and rotation, called the ro-vibrational state. Each such state, identified by quantum numbers describing both the vibration and rotation, has a single energy which depends on said quantum numbers. In the dipole transitions described above, a single photon of radiation is absorbed, transforming the molecule from one ro-vibrational state to another. As the energies of the ro-vibrational states are discrete, so are the energies of the transitions between them.
Therefore, a photon must possess a specific energy to be absorbed by a molecule to transform it between two given ro-vibrational states. Since the energy of a photon is proportional to the frequency of the radiation of which the photon is a part (or equivalently, inversely proportional to the wavelength), only discrete frequencies (wavelengths) can be absorbed by the molecule. The set of discrete frequencies (wavelengths), often called absorption lines, at which a particular species of molecule absorbs, is called the absorption spectrum of said molecule. The width in frequency (wavelength) of each absorption line depends on the specific ro-vibrational transition, the pressure and temperature of the gas containing the molecule, and the presence of other types of molecules in said gas. Each species of molecule has a unique absorption spectrum, by which the species of molecule may be identified. Since the energies of different rotational states of a gaseous molecule are typically spaced much more closely than the energies of different vibrational states, then the absorption lines occur in sets, each set corresponding to a single vibrational transition, and many rotational transitions. These sets of absorption lines are called absorption bands. An instrument which measures an absorption spectrum is called a spectrometer.
FunctionalSpectral RangeSpectral RangeGroup NameBond(μm)(cm−1)HydroxylO—H2.770-2.7473610-3640Aromatic RingC6H63.226-3.3333000-3100AlkeneC═C—H3.247-3.3113020-3080AlkaneC—C—H3.378-3.5092850-2960CarbonylC═O5.714-6.0611650-1750NitrileC≡N4.425-4.5252210-2260Amine IN—H2.857-3.0303300-3500Amine IIC—N7.353-8.4751180-1360
Molecular vibrational bands can be likened to the acoustic frequencies of a string (such as on a violin). Similarly, molecular bands have overtones, which are harmonics of the vibrational motion. The original stretch that produces mid-infrared absorption bands is called the fundamental. A fundamental has many harmonics, as well as combinations of harmonics at a wide variety of frequencies. The absorption at the harmonics is always less than at the fundamental, and can decrease significantly for higher harmonics. Therefore, these overtone transitions are normally called weak overtones.
In the NIR, all the vibrational transitions are harmonics of fundamental, mid-infrared bands. These transitions can be one hundred to ten thousand times weaker than their mid-infrared counterparts. Standard methods, such as Fourier Transform Infrared Spectroscopy (FTIR), commonly used to characterize mid-infrared transitions, often have difficulty detecting these weak absorption features in the NIR spectral region. Therefore, more sensitive detection methods are required to measure NIR absorption features.
Moreover, because overtone bands and combinations of overtone bands often overlap in wavelength (frequency), the NIR is rich with dense bands of absorption lines. It is therefore not uncommon to find spectral regions where the same molecular species has both strong and weak transitions that are co-located in wavelength. Additionally, when it is required to identify one or more species present in a mixture of compounds it is extremely difficult to identify which absorption peak is attributable to a particular molecule.
Measuring the concentration of an absorbing species in a sample is accomplished by applying the Beer-Lambert Law. The Beer-Lambert law (also known as Beer's Law) is the linear relationship between absorbance and concentration of an absorbing species. The Beer-Lambert Law can be derived from an approximation for the absorption of a molecule by considering the molecule as an opaque disk whose cross-sectional area, σ, represents the effective area seen by a photon of frequency ω. If the frequency of the light is far from the transition resonance, the area is approximately 0, and if ω is at resonance the area is a maximum. To derive the absorption in an infinitesimal slab, dz, of a sample as shown in FIG. 1, define the following parameters: Io is the intensity entering the sample at z=0, Iz is the intensity entering the infinitesimal slab at z, dI is the intensity absorbed in the slab, I is the intensity of light leaving the sample, N is the density of absorbing molecules, and B is the cross-sectional area of the radiation. The total opaque area on the slab due to the absorbers is σNB dz. Then, the fraction of photons absorbed will be σNB(dz/B) so that:dI/Iz=−σdz
Integrating this equation from z=0 to z=L, the length of the sample, results in the total transmission, I:ln(I)−ln(Io)=−σNL or −ln(I/Io)=σNL.
By substituting the equation for molar concentration,
C(moles/liter)=N(molecules/cm3)*(1 mole/6.023×1023 molecules)*1000 cm3/liter and the relation between the natural and base 10 logarithms,
2.303*log(x)=ln(x),
the integrated equation becomes:−log(I/Io)=σ(6.023×1020/2.303) CL or −log(I/Io)=A=CεML,where εM=σ(6.023×1020/2.303)=σ2.61×1020 is the molar extinction coefficient.
Typical cross-sections and molar extinction coefficients are:
εMσ (cm2)(M−1 cm−1)Atoms10−123 × 108Molecules10−163 × 104Infrared10−193 × 10Raman scattering10−293 × 10−9
The general Beer-Lambert Law is usually written as:A(λ)=α(λ)L=Cε(λ)L  (1)where A(λ) is the measured absorbance, α(λ) is a wavelength-dependent absorption coefficient, ε(λ) is a wavelength-dependent extinction coefficient, L is the path length, and C is the analyte concentration. When working in concentration units of molarity, the Beer-Lambert Law is written as:A(λ)=αM(λ)L=CεM(λ)L.where αM(λ) is the wavelength-dependent molar absorption coefficient having units of cm−1M−1, and εM(λ) is the wavelength dependent molar extinction coefficient with units of liter cm−1M−1.
Experimental measurements are usually made in terms of transmittance (T), which is defined as:T=I/I0where I is the light intensity immediately after the light passes through the sample and Io is the light intensity immediately before the light impinges on the sample. The relation between A and T is:A=−log T=−log(I/Io)  (2)
A working curve is a plot of the analytical signal (the instrument or detector response) as a function of analyte concentration. These working curves are obtained by measuring the signal from a series of standards of known concentration. The working curves are then used to determine the concentration of an unknown sample or to calibrate the linearity of an analytical instrument.
Modern absorption instruments can usually display the data as transmittance, %-transmittance, or absorbance. An unknown concentration of an analyte can be determined by measuring the amount of light that a sample absorbs and then applying Beer's Law. If the absorption coefficient is not known, the unknown concentration can be determined using a working curve of absorbance versus concentration derived from known standards.
Standards are materials containing a known concentration of a known analyte. They provide a reference to determine unknown concentrations or to calibrate analytical instruments. The accuracy of an analytical measurement is how close a result comes to the true value. Determining the accuracy of a measurement usually requires calibration of the analytical method with a known standard. This is often done with standards of several concentrations to make a calibration or working curve. Standard reference materials are available from standards laboratories such as the National Institute for Standards and Technology (NIST).
The linearity of the Beer-Lambert Law is limited by chemical and instrumental factors. Causes of nonlinearity include:                deviations in absorption coefficients at high concentrations (>0.01M) due to electrostatic interactions between molecules in close proximity        scattering of light due to particulates in the sample        fluorescence or phosphorescence of the sample        changes in refractive index at high analyte concentration        shifts in chemical equilibrium as a function of concentration        non-monochromatic radiation (deviations can be minimized by using a relatively flat part of the absorption spectrum such as the maximum of an absorption band)        stray light        
Equations (1) and (2) show that the ability of a spectrometer to detect a specific concentration depends not only on the path length through the sample, but also on the intensity noise of the light source and detector. Sensitivity can be quantified as a minimum detectable absorption loss (MDAL), i.e., the normalized standard deviation of the smallest detectable change in absorption. MDAL typically has units of cm−1. Sensitivity is defined as the achievable MDAL in a one second measurement interval, and has units of cm−1 Hz−1/2. Sensitivity accounts for the different measurement speeds achieved by diverse absorption-based methods. Sensitivity is a figure of merit for any absorption-based technique.
Typically, a spectral feature (usually called an absorption peak) of the target species is measured in order to obtain its concentration. Although many different species may absorb light at one or more wavelengths, the total spectral profile of any particular species is unique. The ability of a spectrometer to distinguish between two different species absorbing at similar wavelengths is called selectivity. Because spectral features such as absorption peaks become narrow as the sample pressure is reduced, selectivity can be improved by reducing the analyte sample operating pressure. However, the spectrometer must still be able to resolve the resulting spectral lines. Thus, selectivity depends on spectral resolution. Spectral resolution, typically measured in frequency (MHz), wavelength (μm), or wave numbers (cm−1), is another figure of merit for a spectrometer.
Optical detection is the determination of the presence and/or concentration of one or more target species within a sample by illuminating the sample with optical radiation and measuring optical absorption by the sample. A correspondingly wide variety of optical detection methods are known. Examples of such instruments are Fourier Transform Infrared (FTIR), non-dispersive infrared, (NDIR) and tunable diode laser absorption spectroscopy (TDLAS).
None of the above-mentioned existing absorption spectroscopy methods can measure absolute absorption, regardless of whether they utilize incoherent or monochromatic light sources. Therefore, all of these approaches require calibration. Furthermore, these instruments have limited sensitivity because the gas cells they use direct the light through only a limited number of passes.
Cavity enhanced methods resolve the sensitivity limitation by increasing the effective path length. Cavity enhanced optical detection entails the use of a passive optical resonator, also referred to as a cavity. Cavity enhanced absorption spectroscopy (CEAS), integrated cavity output spectroscopy (ICOS) and cavity ring down spectroscopy (CRDS) are three of the most widely used cavity enhanced optical detection techniques. ICOS as used herein is intended to include a recent variant called off-axis ICOS where the light is injected into the resonator at an angle to the optical axis. The teaching of U.S. Pat. Nos. 5,528,040; 5,912,740; 6,795,190 and 6,466,322, which describe these techniques are hereby incorporated herein by this reference.
Cavity ring-down spectroscopy (CRDS) is based on the principle of measuring the rate of decay of light intensity inside a stable optical resonator, called the ring-down cavity (RDC). Once sufficient light is injected into the RDC from a laser source, the input light is interrupted, and the light transmitted by one of the RDC mirrors is monitored using a photodetector. The transmitted light, I(t,λ), from the RDC is given by the equation:I(t,λ)=I0e−t/τ(λ)  (3)where I0 is the transmitted light at the time the light source is shut off, τ(λ) is the ring-down time constant, and R(λ)=1/τ(λ) is the decay rate. The transmitted light intensity decays exponentially over time.
In CRDS, an optical source is usually coupled to the resonator in a mode-matched manner, so that the radiation trapped within the resonator is substantially in a single spatial mode. The coupling between the source and the resonator is then interrupted (e.g., by blocking the source radiation, or by altering the spectral overlap between the source radiation and the excited resonator mode). A detector typically is positioned to receive a portion of the radiation leaking from the resonator, which decays in time exponentially with a time constant τ. The time-dependent signal from this detector is processed to determine τ (e.g., by sampling the detector signal and applying a suitable curve-fitting method to a decaying portion of the sampled signal). Note that CRDS entails an absolute measurement of τ. Both pulsed and continuous wave laser radiation can be used in CRDS with a variety of factors influencing the choice. The articles in the book “Cavity-Ringdown Spectroscopy” by K. W. Busch and M. A. Busch, ACS Symposium Series No. 720, 1999 ISBN 0-8412-3600-3, including the therein cited references, cover most currently reported aspects of CRDS technology.
Single spatial mode excitation of the resonator is also usually employed in CEAS(also called ICOS)) or off-axis ICOS, but CEAS differs from CRDS in that the wavelength of the source is swept (i.e., varied over time), so that the source wavelength coincides briefly with the resonant wavelengths of a succession of resonator modes. A detector is positioned to receive radiation leaking from the resonator, and the signal from the detector is integrated for a time comparable to the time it takes the source wavelength to scan across a sample resonator mode of interest. The resulting detector signal is proportional to τ, so the variation of this signal with source wavelength provides spectral information on the sample. Note that CEAS/ICOS entails a relative measurement of τ. The published Ph.D. dissertation “Cavity Enhanced Absorption Spectroscopy”, R. Peeters, Katholieke Universiteit Nijmegen, The Netherlands, 2001, ISBN 90-9014628-8, provides further information on both CEAS and CRDS technology and applications CEAS is discussed in a recent article entitled “Incoherent Broad-band Cavity-enhanced Absorption Spectroscopy by S. Fiedler, A. Hese and A, Ruth Chemical Physics Letters 371 (2003) 284-294. The teaching of U.S. Pat. No. 6,795,190 which describes ICOS and off-axis ICOS are incorporated herein.
In cavity enhanced optical detection, the measured ring-down time depends on the total round trip loss within the optical resonator. Absorption and/or scattering by target species within the cavity normally accounts for the major portion of the total round trip loss, while parasitic loss (e.g., mirror losses and reflections from intracavity interfaces) accounts for the remainder of the total round trip loss. The sensitivity of cavity enhanced optical detection improves as the parasitic loss is decreased, since the total round trip loss depends more sensitively on the target species concentration as the parasitic loss is decreased. Accordingly, both the use of mirrors with very low loss (i.e., a reflectivity greater than 99.99 percent), and the minimization of intracavity interface reflections are important for cavity enhanced optical detection. Although the present invention will be described primarily in the context of CRDS, it should be understood that the methodology is also applicable to CEAS including ICOS and off-axis ICOS.
In the study of molecular absorption lines one must be aware that the radiation induced atomic or molecular energy level transitions are not precisely “sharp”, i.e., they are not “delta functions” in wavelength. There is always a finite width to the observed spectral lines. The shape of a spectral line is described by the line profile function and its width by its full width at half maximum intensity (peak intensity) (FWHM). The physical line shape is due to the combined effects of the different broadening processes. The physical shape described by the Voigt function is known as the Voigt profile and takes into account natural broadening, Doppler broadening and pressure broadening. Line shift is the displacement of the central wavelength of the spectral line due to similar effects.
Natural broadening has its origin in the finite optical lifetime of one or both of the levels involved in a transition. Doppler broadening is due to the random motion of the emitting or absorbing atoms. A Doppler broadened line has a Gaussian shape. A Doppler shift is a line shift caused by the Doppler effect. Collisional broadening and collisional shift of the line is produced by collisions of the emitting or absorbing particle with other particles. When collisions occur between unlike, neutral particles, the term foreign-gas broadening is appropriate. When the colliding particles are of the same species, one uses the term resonance broadening. When collisions take place with charged particles or particles with a strong permanent electrical dipole moment, the term is Stark broadening. A strong chaotic electrical field causes Stark broadening, whereas an applied static electrical field induces a Stark shift.
One source of broadening is the natural line width, which arises from the uncertainty in the energy of the states involved in the transition. The Heisenberg Uncertainty Principle suggests that for particles with extremely short lifetimes, there will be a significant uncertainty in the measured energy. The numerous repeated measurements of the mass energy of an unstable particle produces a distribution of energies having a Lorentzian, or Breit-Wigner, distribution. A Lorentzian distribution resembles a Gaussian distribution near the peak, but its tails are much flatter.
If the width of this distribution at half-maximum, or full width half maximum (FWHM), is labeled Γ, then the uncertainty in energy ΔE can be expressed as:
      Δ    ⁢                  ⁢    E    =            Γ      2        =          ℏ              2        ⁢        ζ            where the particle lifetime ζ is taken as the uncertainty in time, ζ=Δt.
Γ is often referred to as the “natural line width”. For optical spectroscopy it is a minor factor because the natural linewidth is typically about 5×10−4 cm−1 (or 10−8 seconds or 15 MHz), a tenth as much as the Doppler broadening. At atmospheric pressures, the lines are dominated by pressure broadening. At lower pressures the dominant broadening mechanism transitions from pressure to Doppler. The natural linewidth dominates the broadening only at very low temperature and pressure.
For molecular spectra in the infrared, the limit on spectral resolution is often set by Doppler broadening. With the thermal motion of the atoms, those atoms traveling toward the detector with a velocity v will have transition frequencies which differ from those of atoms at rest by the Doppler shift, as shown in FIG. 3. The distribution can be found from the Boltzmann distribution. From the Doppler shifted wavelength, the observed frequency is:
  v  =            c              λ        ″              =                            v          0                ⁢                              1            -                                          v                s                2                                            c                2                                                                (                  1          -                                    v              s                        c                          )            
Rearranging gives the more convenient form:
  v  =            v      0        ⁢                            1          +                                    v              s                        c                                    1          -                                    v              s                        c                              where the relative velocity νs is positive if the source is approaching and negative if receding.
The Maxwell speed distribution for the molecules of an ideal gas is shown in FIG. 4. Since the thermal velocities are non-relativistic, the Doppler shift in the angular frequency is given by the simple form:
                    ω        =                              ω            0                    ⁡                      (                          1              ±                              v                c                                      )                                                        ω          0                =                  frequency  for  an  atom  at  rest                    
From the Boltzmann distribution, the number of atoms with velocity v in the direction of the observed light is given by:
            n      ⁡              (        v        )              ⁢    dv    =      N    ⁢                            m          0                          2          ⁢          π          ⁢                                          ⁢          kT                      ⁢          ⅇ                        -                      m            0                          ⁢                              v            2                    /          2                ⁢        kT              ⁢    dv  where N is the total number of molecules, m0 is the molecular mass, and k is Bolzmann's constant. The distribution of radiation around the center frequency is then given by:
      I    ⁡          (      ω      )        =            I      0        ⁢          exp      ⁡              [                                            -                              m                0                                      ⁢                                                            c                  2                                ⁡                                  (                                                            ω                      0                                        -                    ω                                    )                                            2                                            2            ⁢                          kTω              0              2                                      ]            
This is in the form of a Gaussian, and the FWHM is given by:
      Δ    ⁢                  ⁢          ω      Doppler        =                    2        ⁢                  ω          0                    c        ⁢                  2        ⁢        ln        ⁢                                  ⁢        2        ⁢                  kT                      m            0                              
Often it is convenient to express this in terms of wavelength:
      Δλ          λ      0        =      2    ⁢                  2        ⁢        ln        ⁢                                  ⁢        2        ⁢                  kT                                    m              0                        ⁢                          c              2                                          
Note that Doppler broadening is independent of the pressure and depends only on the molecular weight and the temperature of the molecules being measured. Doppler linewidths for different gases in gaseous backgrounds can range from tens of MHz to several hundred MHz.
When one moves further to longer wavelengths into the microwave region for molecular rotational spectra, the natural line width again emerges as a larger source of broadening than Doppler broadening. For infrared vibrational transitions, however, there is another form of line shape broadening that dominates at atmopheric pressure (so called “pressure broadening”).
In a sample of gas, the molecules of the species of interest will be continuously colliding with themselves, as well as with molecules of the background matrix. These collisions reduce the natural lifetime, and through the uncertainty principle increase the uncertainty in energy (or spectral width). This pressure broadening results from the perturbations of rotational energy levels by molecular collisions, and can be viewed as the overlapping of the potential fields of two molecules. This type of spectral line shape broadening increases linearly with the collision rate (and thus, to a first order, with pressure) because it depends on the number of collisions per second, i.e., on the number density of the molecules. It also depends on the relative speed of the molecules, and therefore depends on the square root of the temperature. The resulting line shape is Lorentzian. The distribution of radiation around the center frequency is then given by:
      I    ⁡          (      ω      )        =            I      0        ⁢                  2        ⁢                  π          /          Δ                ⁢                                  ⁢                  ω          L                                                  (                                          ω                -                                  ω                  0                                                            Δ                ⁢                                                                  ⁢                                  ω                  L                                                      )                    2                +                  π          2                    
Where ΔωL is the pressure broadened FWHM and ω0 is the line center. As is shown in FIG. 2, the Lorentzian line shape has wings which extend much farther than for a Gaussian line shape, and it can produce significant absorption far from line center.
The Lorentzian linewidth is in general found from:
      Δ    ⁢                  ⁢          ω      L        =      Δ    ⁢                  ⁢                  ω        L            ⁡              (        STP        )              ⁢                  P        ⁢                  T                                      P          L                ⁢                              T            L                              where PL is standard pressure (1 atm), TL is standard temperature (296° K), ΔωL(STP) is the Lorentzian line width at Standard Temperature and Pressure, P is the molecular number density (or pressure) and T is the sample temperature. For a system operating at constant sample temperature, the Lorentzian line width shape is simply proportional to pressure.
Note, however, that in all cases, the strength of a given transition is intrinsic to the molecule, so that the integrated intensity is conserved. If the sample is dilute in the matrix background gas, then ΔωL will depend on total pressure, and the broadening will be called simply “collisional broadening”. However, when the collisions of the target gas are primarily with themselves, ΔωL is primarily determined by the partial pressure of the target gas, and the broadening is called “self broadening”.
FIGS. 5a and 5b show the self-broadened and pressure-broadened FWHM as a function of pressure for the first overtone of HCl at 1742 nm (5577.33 cm−1) [M De Rosa et al., Applied Physics B, vol. 72, p. 245-248 (2001)]. At constant temperature, note the linear dependence. Also note that for a typical Doppler broadening FWHM of 350 MHz at room temperature, the pressure broadening becomes comparable to the Doppler broadening at 40 Torr. From FIG. 5b it is clear that pressure broadening can strongly depend on the particular background gas. The interactions are significantly stronger for N2 and O2 than for Ar and He. HCl is a molecule with a large dipole moment, a quadrupole moment, and polarizabilty. While N2 and O2 have no dipole moment, they do have a quadrupole moment and also have a finite polarizability. The interaction is stronger for N2, which has a larger quadrupole moment than O2. Furthermore, the interaction of N2 and O2 with HCl is stronger than that of Ar and He, which have no dipole or quadrupole moment, and small polarizability. The polarizability for Ar is comparable to that of O2, so that the interaction in pressure broadening is similar. However, He has a much smaller (8 times) polarizability, resulting in a much weaker pressure broadening dependence.
In general, the overall spectral lineshape at atmospheric pressure is a convolution of a Lorenztian and Doppler profile, reflecting the contributions of both pressure and Doppler broadening mechanisms. This convolution is call a Voigt profile:
            σ      n        ⁡          (      v      )        =      S    ⁢          a                        π                      3            /            2                          ⁢                  α          D                      ⁢                  ∫                  -          ∞                          +          ∞                    ⁢                                    ⅇ                          -                              y                2                                              ⁢                      ⅆ            y                                                              (                                                δ                  ⁢                                                                          ⁢                  v                                -                y                            )                        2                    +                      a            2                              where α=ΔωL/ΔωD is the damping ratio and δν=(ν−ν0)/ΔωD.
For small damping ratios, a<<1 the line shape becomes primarily Doppler broadened, while for a>1, the line shape becomes primarily Lorentzian. In general, the Voight profile adopts a Doppler-like behavior in the line center, and a Lorentz-like behavior in the line wings. The Voight profile must be evaluated by numerical integration.
By computing the change in the Voigt profile, for example for water vapor at 1392 nm, researchers have been able to estimate that the peak absorption typically remains constant when pressure broadening is the dominant lineshape broadening mechanism. For water, the pressure broadening coefficient is linear with pressure, but the peak absorption stays almost constant between atmospheric pressure and 100 Torr (it drops only 4% in the peak although the number density of water decreases by a factor of 7.6). This is illustrated in FIGS. 6a and 6b (W. J. Kessler et al., SPIE Paper No. 3537-A30 (1998) the teaching of which is incorporated herein by this reference.